In recent times, multi-antenna (or multiple-input and multiple-output, MIMO) systems have been adopted to increase throughput, known as multiplexing gain, and improve reliability, known as diversity gain, in wireless communication. In single user MIMO (SU-MIMO)—a single transmitter (TX) equipped with multiple antennas transmits a certain number of streams to a single receiver (RX) equipped with multiple antennas. Each signal transmitted from the transmit antennas will be received, after been filtered by the channel, by all the receiving antennas, thus each receiving antenna receives a scattered and interfered version of each of the transmit symbols.
Several techniques have been created in order to mitigate negative interference effects, or, more smartly, to exploit this interference, in order to improve the detection of the transmitted signal. Remarkably, under the assumption of Gaussian signaling, perfect channel knowledge at the TX and the RX, perfect power and bit allocation at the transmitter, a simple linear precoder and decoder obtained by the singular value decomposition (SVD) of the channel matrix is sufficient to achieve optimal performance. However, there exist a number of cases in which those assumptions cannot be respected. For instance, real transmissions do not have Gaussian signaling properties, they always being drawn from some finite size alphabet, generally M-QAM or M-PSK. Secondly, in many cases the TX is not controllable by the RX, or it is oblivious of the channel matrix. In these cases, there arises the problem of how the RX can minimize its bit error rate (BER) by employing some smart equalization algorithm.
So far, the most effective algorithm, from this point of view, is the maximum likelihood (ML) approach, which compares all possible transmitted signals filtered by the channel matrix with the received signal and selects the one with the minimum distance. This approach is clearly impractical, since it requires an enormously high amount of calculation, since the number of operation grows exponentially with the constellation size and the number of transmit antennas.
Henceforth, a number of algorithms have been developed, most remarkably the sphere decoder (SD), which searches a reduced ML space with lower complexity and efficiency, the MMSE-successive interference cancellation (MMSE-SIC), which once it decodes a symbol, it subtracts the interference it creates to other symbols prior decoding them, and linear technique such as minimum mean square error (MMSE) and zero-forcing (ZF), which decode by multiplying the received vector by a particularly designed decoding matrix.
Each approach has its own advantages and disadvantages. Linear solutions have a low complexity and have almost optimal performance at low level of Eb/N0. SD is close to optimal, but its already high complexity grows with the constellation size of the employed communication scheme. MMSE-SIC has low complexity, but it is anyway wasted at low Eb/N0.
Consider a single user point-to-point MIMO link as the one depicted in FIG. 1. A transmitter comprises a channel coder 3 and a MIMO precoder 4. A receiver 2 comprises a MIMO equalizer 5 and a channel decoder 6. The transmitter 1 employs a sub-optimal precoding strategy known at the receiver 2. The receiver 2 needs to minimize its own BER using the minimum possible complexity. The transmitter 1 is equipped with Nt transmit antennas, the receiver 2 with NRx receive antennas, yielding a channel matrix {tilde over (H)}∈NRx×Nt.
The transmitter 1 shapes its transmit vector by means of a linear precoder W. From the receiver point of view, this transforms the channel matrix into an effective channel matrix H=HW, with H∈NRx×Nt. The transmit vector is denoted by x=[x1, x2, . . . xNt]T, with each element of the vector belonging to a finite size constellation such as BPSK or 16-QAM. The dimension of the constellation is indicated by the letter M. The received signal is denoted by y=[y1, y2, . . . yNRx]T, where y=Hx+n. Here, n is modeled as a complex Gaussian additive noise of which each entry has variance σ2. The goal of an equalization algorithm is to give an estimation as precise as possible of the vector x knowing y and H. This estimation can be expressed as:x=gH(y)where the function gH(⋅) is the equalization function, indexed from the channel matrix H. The hard-decoding function is denoted by fSTEP(⋅).
To equalize MIMO channels, there exist in the literature a number of solutions. In general, these solutions can be divided into two large sets depending on the nature of the function gH(⋅): linear and non-linear equalizations.
Linear solutions consider the interference as Gaussian noise and they attempt to mitigate its negative effects. Simply, they consist in pre-multiplying the received vector for a matrix, the so called decoder, prior to applying the hard decoding.
Non-linear solutions are more complex algorithms and they exploit the peculiar nature of the interference in order to improve the equalization performance.
Linear solutions, such as ZF and MMSE, have a low complexity level and, compared to ML equalization, show good performance at low Eb/N0 level, whereas they lose a significant amount of information at medium and high Eb/N0. ML solutions maximize the performance of the equalizer but are characterized by high, sometimes unfeasible, complexity. Almost ML solutions, such as SD, reduce the search space of the ML but are anyway unfit for low complexity device and high order modulations. MMSE successive interference cancellation—MMSE-SIC—balances between the performance of the ML and the complexity of MMSE, having as an extra feature the fact of a complexity that is almost independent from the constellation size. However, its complexity is not justified when at low SNR.
Algo-Perfor-rithmmanceComplexityNotesMLOptimal≈ NtNRxMNrSDAlmost≈ Nt3 + Nt2 + M(Nt2 + Nt) + 2NtOptimisticoptimallower(variable)bound,onlyavailable forNt = NRx MMSE- SICVery High  ≈                    N        t            ⁢              N        Rx              +          N      Rx        +                  3        2            ⁢              (                              N            Rx            2                    +                      N            Rx                          )             MMSE-Low at≈ NtNRx + NRxLinearhigh SNR
Therefore, always a trade-off between complexity and accuracy of the different decoding approaches has to be taken into account. At present there exists no decoding approach, which can adaptively handle different situations at optimal complexity and accuracy.